Two-point active microrheology in a viscous medium exploiting a motional resonance excited in dual-trap optical tweezers

Two-point microrheology measurements from widely separated colloidal particles approach the bulk viscosity of the host medium more reliably than corresponding single-point measurements. In addition, active microrheology offers the advantage of enhanced signal to noise over passive techniques. Recently, we reported the observation of a motional resonance induced in a probe particle in dual-trap optical tweezers when the control particle was driven externally [Paul et al., Phys. Rev. E 96, 050102(R) (2017)]. We now demonstrate that the amplitude and phase characteristics of the motional resonance can be used as a sensitive tool for active two-point microrheology to measure the viscosity of a viscous fluid. Thus, we measure the viscosity of viscous liquids from both the amplitude and phase response of the resonance, and demonstrate that the zero crossing of the phase response of the probe particle with respect to the external drive is superior compared to the amplitude response in measuring viscosity at large particle separations. We compare our viscosity measurements with those using a commercial rheometer and obtain an agreement $\sim 1\%$. The method can be extended to viscoelastic material where the frequency dependence of the resonance may provide further accuracy for active microrheological measurements.

Figure 1

(a) Resonance frequency as a function of viscosity for different stiffnesses. ${\mathit{k}}_{\mathit{2}}$ is kept constant (at 3 ${\displaystyle \mu \text{N}}$/m), while ${\mathit{k}}_{\mathit{1}}$ is varied. We use ${\mathit{k}}_{\mathit{1}}$ values of 7 (left plot, thick line, red), 15 (center plot, thin line, green), and 30 (right plot, dashed line, blue) ${\displaystyle \mu \text{N}}$/m, respectively. (b) Amplitude of probe as a function of drive frequency for same stiffness values and color code as (a). The viscosity value was constant at $850\mu \mathrm{Pa}\mathrm{s}$ and the amplitude of the driving signal at 340 nm. (c) Phase of probe as a function of drive frequency [color code same as (a)]. (d) Amplitude of probe as a function of drive frequency for stiffness $k_2=3{\displaystyle \mu \text{N}}$/m, $k_1=7{\displaystyle \mu \text{N}}$/m and different viscosity ($\eta$) values of 100 (left plot, dashed line, blue), 850 (center plot, thin line, green), and 5000 (right plot, thick line, red) $\mu \mathrm{Pa}\mathrm{s}$.

Figure 2

Phase (left axis, filled markers with solid lines) and amplitude (right axis, filled markers with dashed lines) sensitivities as a function of (a) $k_1$ and (b) viscosity. Once again, ${\mathit{k}}_{\mathit{2}}$ is kept constant (at $3{\displaystyle \mu \text{N}}$/m), while ${\mathit{k}}_{\mathit{1}}$ is varied. In (a) the red (top), green (center), and blue (bottom) lines signify viscosity values of 100, 850, and $5000\mu \mathrm{Pa}\mathrm{s}$ respectively. In (b) the red (top), green (center), and blue (bottom) lines signify ${\mathit{k}}_{\mathit{1}}$ values of 7, 15, and $30{\displaystyle \mu \text{N}}$/m, respectively. In the inset, a zoomed-in version of the phase and amplitude sensitivities for large viscosity values is shown.

Figure 3

Schematic of the experimental setup. $\frac{\lambda }{2}$: half-wave plate, L: lens, M: mirror, AOM: acousto-optical modulator, PBS: polarizing beam splitter, DC: dichroic mirror, E: edge mirror, P: polarizer PA, PB: photodiodes (Thorlabs PDA100A-EC), PC: personal computer.

Figure 4

(a) The control and probe particles trapped in different $z$ planes, which is apparent from the different sizes. (b) The particles trapped in the same plane.

Figure 5

(a), (c) Show the position probability density function plotted against particle displacement for the control particle for trap separations of 9 and $4\mu \text{m}$, respectively, while (b) and (d) show that for the probe particle for the same separations. The solid circles in black are the probability values computed from experimentally measured data, which are then fit Gaussian functions (solid black lines). Each figure also shows the corresponding trap potentials calculated from the same experimental data (right axis, open gray circles), which are fit to quadratic functions of the displacement (solid gray lines).

Figure 6

Simulated intensity gradient using our trap parameters. The force acting on the trapped particle is proportional to the intensity gradient, the linear region of which appears to be about $0.3\mu \text{m}$ from a straight line fit around the center.

Figure 7

Experimentally measured amplitude responses of the driven particle for two different viscosity solutions. The experimentally measured data are shown in red filled triangles and blue filled squares for viscosities of 890 and $2000\mu \mathrm{Pa}\mathrm{s}$, respectively. The data are fit to the amplitude response given in Eq. (12) viscosities of 890 (right plot, dashed line, red) and 2000 (left plot, solid line, blue) $\mu \mathrm{Pa}\mathrm{s}$, respectively.